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  4. If function f(x) is differentiable at x = a, then displaystyle limx→ a (x2f(a)-a2f(x)/x-a) is :

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Q. If function f(x) is differentiable at x = a, then $\displaystyle \lim_{x\to a} \frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$ is :

AIEEEAIEEE 2011Continuity and Differentiability

Solution:

$\displaystyle \lim_{x\to a} \frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$= \displaystyle \lim_{x\to a} \frac{2xf\left(a\right)-a^{2}f'\left(x\right)}{1}$
$= 2af\left(a\right) - a^{2}f' \left(a\right)$
Alter$\quad \displaystyle \lim_{x\to a} \frac{x^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$= \displaystyle \lim_{x\to a} \frac{x^{2}f\left(a\right)-a^{2}f\left(a\right)+a^{2}f\left(a\right)-a^{2}f\left(x\right)}{x-a}$
$= \displaystyle \lim_{x\to a} \frac{\left(x^{2}-a^{2}\right)f\left(a\right)+a^{2}\left(f\left(x\right)-f\left(a\right)\right)}{x-a}$
$= \displaystyle \lim_{x\to a} \left(x + a\right) f\left(a\right) - a^{2} \left\{\frac{f\left(x\right) - f\left(a\right)}{\left(x-a\right)}\right\}$
$= 2af\left(a\right) - a^{2}f' \left(a\right)$